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Hamiltonian Mechanics Hamiltonian Mechanics December 5, 2012 1 Phase space Phase space is a dynamical arena for classical mechanics in which the number of independent dynamical variables is doubled from n variables qi, i = 1; 2; : : : ; n to 2n by treating either the velocities or the momenta as independent variables. This has two important consequences. First, the equations of motion become first order differential equations instead of second order, so that the initial conditions is enough to specify a unique point in phase space. The means that, unlike the configurations space treatment, there is a unique solution to the equtions of motion through each point. This permits some useful geometric techniques in the study of the system. Second, as we shall see, the set of transformations that preserve the equations of motion is enlarged. In Lagrangian mechanics, we are free to use n general coordinates, qi, for our description. In phase space, however, we have 2n coordinates. Even though transformations among these 2n coordinates are not com- pletely arbitrary, there are far more allowed transformations. This large set of transformations allows us, in principal, to formulate a general solution to mechanical problems via the Hamilton-Jacobi equation. 1.1 Velocity phase space While we will not be using velocity phase space here, it provides some motivation for our developments in the next Sections. The formal presentation of Hamiltonian dynamics begins in Section 1.3. Suppose we have an action functional S = L (q ; q_ ; t) dt ˆ i j dependent on n dynamical variables, qi (t), and their time derivatives. We might instead treat L (qi; uj; t) as a function of 2n dynamical variables. Thus, instead of treating the the velocities as time derivatives of the position variables, (qi; q_i) we introduce n velocities ui and treat them as independent. Then the variations of the velocities δui are also independent, and we end up with 2n equations. We then include n constraints, restoring the relationship between qi and q_i, h X i S = L (q ; u ; t) + λ (_q − u ) dt ˆ i j i i i Variation of the original dynamical variables then results in 0 = δqS @L = δqi + λiδq_i dt ˆ @qi @L _ = − λi δqidt ˆ @qi 1 so that @L _ λi = @qi For the velocities, we find 0 = δuS @L = − λi δuidt ˆ @ui so that @L λi = @ui and finally, varying the Lagrange multipliers, λi, we recover the constraints, ui =q _i We may eliminate the multipliers by differentiating the velocity equation d d @L (λi) = dt dt @ui _ _ to find λi, then substituting for ui and λi into the qi equation, d @L _ λi = dt @q_i d @L @L = dt @q_i @qi d @L @L − = 0 dt @q_i @qi Pn 1 2 and we recover the Euler-Lagrange equations. If the kinetic energy is of the form i=1 2 mui , then the Lagrange multipliers are just the momenta, @L λi = @ui = mui = mq_i 1.2 Phase space We can make the construction above more general by requiring the Lagrange multipliers to always be the conjugate momentum. Combining the constraint equation with the equation for λi we have @L λi = @q_i We now define the conjugate momentum to be exactly this derivative, @L pi ≡ @q_i Then the action becomes h X X i S = L (q ; u ; t) + p q_ − p u dt ˆ i j i i i i h X X i = L (q ; u ; t) − p u + p q_ dt ˆ i j i i i i 2 For Lagrangians quadratic in the velocities, the first two terms become X X L (qi; uj; t) − piui = L (qi; q;_ t) − piq_i X = T − V − piq_i = − (T + V ) We define this quantity to be the Hamiltonian, X H ≡ piq_i − L (qi; q;_ t) Then hX i S = p q_ − H dt ˆ i i This successfully eliminates the Lagrange multipliers from the formulation. The term “phase space” is generally reserved for momentum phase space, spanned by coordinates qi; pj. 1.2.1 Legendre transformation Notice that H is, by definition, independent of the velocities, since X H = pjq_j − L 0 1 @H @ X = p q_ − L @q_ @q_ @ j j A i i j X @L = p δ − j ij @q_ j i @L = pi − @q_i ≡ 0 Therefore, the Hamiltonian is a function of qi and pi only. This is an example of a general technique called Legendre transformation. Suppose we have a function f, which depends on independent variables A; B and dependent variables, having partial derivatives @f = P @A @f = Q @B Then the differential of f is df = P dA + QdB A Legendre transformation allows us to interchange variables to make either P or Q or both into the independent variables. For example, let g (A; B; P ) ≡ f − PA. Then dg = df − AdP − P dA = P dA + QdB − AdP − P dA = QdB − AdP 3 so that g actually only changes with B and P , g = g (B; P ). Similarly, h = f − QB is a function of (A; Q) only, while k = − (f − PA − QB) has (P; Q) as independent variables. Explicitly, dk = −df + P dA + AdP + QdB + BdQ = AdP + BdQ and we now have @f = A @P @f = B @Q 2 Hamilton’s equations The essential formalism of Hamilton’s equation is as follows. We begin with the action S = L (q ; q_ ; t) dt ˆ i j and define the conjugate momenta @L pi ≡ @q_i and Hamiltonian X H (qi; pj; t) ≡ pjq_j − L (qi; q_j; t) Then the action may be written as hX i S = p q_ − H (q ; p ; t) dt ˆ j j i j where qi and pj are now treated as independent variables. Finding extrema of the action with respect to all 2n variables, we find: 0 = δqk S P @ j pjq_j @H = δq_k − δqk dt ˆ @q_k @qk 0 1 X @H = p δ δq_ − δq dt ˆ @ j jk k @q kA j k @H = pkδq_k − δqk dt ˆ @qk @H = −p_k − δqkdt ˆ @qk so that @H p_k = − @qk and 0 = δpkS P @ j pjq_j @H = δpk − δpk dt ˆ @pk @pk 4 @H = q_kδpk − δpk dt ˆ @pk @H = q_k − δpkdt ˆ @pk so that @H q_k = @pk These are Hamilton’s equations. Whenever the Legendre transformation between L and H and between q_k and pk is non-degenerate, Hamilton’s equations, @H q_k = @pk @H p_k = − @qk form a system equivalent to the Euler-Lagrange or Newtonian equations. 2.1 Example: Newton’s second law Suppose the Lagrangian takes the form 1 L = mx_ 2 − V (x) 2 Then the conjugate momenta are @L pi = @x_ i = mx_ i and the Hamiltonian becomes X H (xi; pj; t) ≡ pjx_ j − L (xi; x_ j; t) 1 = mx_ 2 − mx_ 2 + V (x) 2 1 = mx_ 2 + V (x) 2 1 = p2 + V (x) 2m Notice that we must invert the relationship between the momenta and the velocities, p x_ = i i m then expicitly replace all occurrences of the velocity with appropriate combinations of the momentum. Hamilton’s equations are: @H x_ k = @pk p = k m @H p_k = − @xk @V = − @xk thereby reproducing the usual definition of momentum and Newton’s second law. 5 2.2 Example Suppose we have a coupled oscillator comprised of two identical pendula of length l and each of mass m, connected by a light spring with spring constant k. Then for small displacements, the action is 1 1 S = ml2 θ_2 + θ_2 − k (l sin θ − l sin θ )2 − mgl (1 − cos θ ) − mgl (1 − cos θ ) dt ˆ 2 1 2 2 1 2 1 2 which for small angles becomes approximately 1 1 1 S = ml2 θ_2 + θ_2 − k (lθ − lθ )2 − mgl θ2 + θ2 dt ˆ 2 1 2 2 1 2 2 1 2 The conjugate momenta are: @L p1 = _ @θ1 2 _ = ml θ1 @L p2 = _ @θ2 2 _ = ml θ2 and the Hamiltonian is _ _ H = p1θ1 + p2θ2 − L 1 1 1 = ml2θ_2 + ml2θ_2 − ml2 θ_2 + θ_2 − k (lθ − lθ )2 − mgl θ2 + θ2 1 2 2 1 2 2 1 2 2 1 2 1 1 1 = ml2 θ_2 + θ_2 + k (lθ − lθ )2 + mgl θ2 + θ2 2 1 2 2 1 2 2 1 2 1 1 1 = p2 + p2 + k (lθ − lθ )2 + mgl θ2 + θ2 2ml2 1 2 2 1 2 2 1 2 Notice that we always eliminate the velocities and write the Hamiltonian as a function of the momenta, pi. Hamilton’s equations are: @H _ θ1 = @p1 1 = p ml2 1 @H _ θ2 = @p2 1 = p ml2 2 @H p_1 = − @θ1 1 = − kl2 (θ − θ ) − mglθ 2 1 2 1 @H p_2 = − @θ2 1 = kl2 (θ − θ ) − mglθ 2 1 2 2 6 _ From here we may solve in any way that suggests itself. If we differentiate θ1 again, and use the third equation, we have 1 θ¨ = p_ 1 ml2 1 1 1 g = − kl2 (θ − θ ) − θ ml2 2 1 2 l 1 k 1 g = − (θ − θ ) − θ m 2 1 2 l 1 Similarly, for θ2 we have k 1 g θ¨ = (θ − θ ) − θ 2 m 2 1 2 l 2 Subtracting, k 1 k 1 θ¨ − θ¨ = − (θ − θ ) − (θ − θ ) 1 2 m 2 1 2 m 2 1 2 d2 k (θ − θ ) + (θ − θ ) = 0 dt2 1 2 m 1 2 so that r r k k θ − θ = A sin t + B cos t 1 2 m m Adding instead, we find g θ¨ + θ¨ = − (θ + θ ) 1 2 l 1 2 so that rg rg θ + θ = C sin t + D cos t 1 2 l l 3 Formal developments In order to fully appreciate the power and uses of Hamiltonian mechanics, we make some formal developments.
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